       Re: Mind+Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg78085] Re: [mg78010] Mind+Mathematica
• From: DrMajorBob <drmajorbob at bigfoot.com>
• Date: Fri, 22 Jun 2007 06:48:03 -0400 (EDT)
• References: <28708894.1182424059011.JavaMail.root@m35>
• Reply-to: drmajorbob at bigfoot.com

```Or (in v6):

step1 = Sin[z]*Sin[z^3 + z] // TrigExpand // Expand

Cos[z^3]/2 - 1/2 Cos[z]^2 Cos[z^3] + 1/2 Cos[z^3] Sin[z]^2 +
Cos[z] Sin[z] Sin[z^3]

step2 = Integrate[#, {z, 0, Infinity}] & /@ step1

(\[Pi] (AiryAi[-2/3^(1/3)] - AiryAi[2/3^(1/3)]))/(4 3^(1/3)) +
Gamma[1/3]/(
4 Sqrt) + (-3^(
1/6) \[Pi] (AiryAi[-2/3^(1/3)] + AiryAi[2/3^(1/3)]) +
Gamma[1/3])/(
8 Sqrt) + (\[Pi] (-6 +
3^(2/3) (AiryAi[-2/3^(1/3)] + AiryAi[2/3^(1/3)]) Gamma[-1/3]))/(
72 Gamma[2/3])

step3 = step2 // FullSimplify

1/6 \[Pi] (-3^(2/3) AiryAi[2/3^(1/3)] - 3/Gamma[-1/3])

step3 // N

0.295741

Bobby

On Thu, 21 Jun 2007 04:45:59 -0500, dimitris <dimmechan at yahoo.com> wrote:

> The integral
>
> Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}]
>
> (as I was informed)
>
> gives a incorrectly divergent message.
> The integral however is convergent.
>
> The following is part of my response to another forum.
> Demonstrate how vital is to help Mathematica sometimes.
>
> In:=
> \$Version
>
> Out=
> "5.2 for Microsoft Windows (June 20, 2005)"
>
> In:=
> int=Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}](*the integral
> stays unevaluated*)
>
> Out=
> Integrate[Sin[z]*Sin[z + z^3], {z, 0, Infinity}]
>
> In:=
> int2 = (int /. Integrate[f_, x_] :> Integrate[#1, {z, 0, Infinity}]
> & ) /@ Expand[Sin[z]*TrigExpand[Sin[z^3 + z]]]
>
> Out=
> (1/72)*(2*Sqrt*Pi*(BesselI[1/3, (4*Sqrt[2/3])/3] - BesselJ[1/3,
> (4*Sqrt[2/3])/3]) +
>     3*Gamma[1/3]*(2*Sqrt - Sqrt*BesselI[-(1/3), (4*Sqrt[2/3])/
> 3]*Gamma[2/3] - Sqrt*BesselJ[-(1/3), (4*Sqrt[2/3])/3]*Gamma[2/3]))
> + Integrate[Cos[z]*Sin[z]*Sin[z^3], {z, 0, Infinity}]
>
> In:=
> int3 = (1/2)*Integrate[Sin[2*z]*Sin[z^3], {z, 0, Infinity}]
>
> Out=
> (Pi*(AiryAi[-(2/3^(1/3))] - AiryAi[2/3^(1/3)]))/(4*3^(1/3))
>
> In:=
> FullSimplify[int2 /. Integrate[x___] :> int3]
>
> Out=
> (-2*3^(1/6)*Pi*AiryAi[2/3^(1/3)] + Gamma[1/3])/(4*Sqrt)
>
> In:=
> N[%, 40]
>
> Out=
> 0.295741225849781931593673891336119670357883693300484102195`40.
>
> Brought to you by M^2
> (Man+Mathematica!)
>
> Dimitris
>
> PS
> I spent almost two hours to figure out a workaround.
> How ancient Greeks said:
> "It is not easy to get Goods"
>
> PS2
> Enjoy Mathematics and Mathematica!
>
>
>

-- =

DrMajorBob at bigfoot.com

```

• Prev by Date: Re: v5.2 preferred for stability over v6.0
• Next by Date: Re: My problem when solving a system of equations
• Previous by thread: Re: Mind+Mathematica
• Next by thread: Solving a Integral